## Quantum Effects in Carbon Nanotubes

STM can probe the transition from 1D delocalized states to molecular levels since voltage pulses can first be used to systematically cut nanotubes into short lengths [29,46] (Fig. 9). Subsequently, these finite-size nanotubes can be characterized spectroscopically.

Venema et al. [47] first reported investigations of quantum size effects in a ~ 30 nm metallic, armchair nanotube shortened by STM voltage pulses. Current vs. voltage measurements carried out near the middle of the tube showed an irregular step-like behavior. The steps in the spectra observed over a small voltage range (±0.2 V) correspond to quantized energy levels entering the bias window as the voltage is increased, and the irregularity in the step spacing is due to coulomb charging effects competing with the 1D level spacing. Remarkably, they discovered that compilation of 100 consecutive I-V measurements spaced 23 pm apart into a spectroscopic map exhibited dI/dV peaks which varied periodically with position along the tube axis (Fig. 10a). This periodic variation in dl/dV as a function of position along the tube, 0.4 nm,

Position along the tube axis, x mm)

Fig. 10. STM line scans. (a) Spectroscopic image compiled from 100 dl/dV measurements. The periodicity is determined from the square of the amplitude of the electron wavefunction at discrete energies. (b) Topographic line profile of atomic corrugation in a shortened armchair nanotube [47]

Position along the tube axis, x mm)

Fig. 10. STM line scans. (a) Spectroscopic image compiled from 100 dl/dV measurements. The periodicity is determined from the square of the amplitude of the electron wavefunction at discrete energies. (b) Topographic line profile of atomic corrugation in a shortened armchair nanotube [47]

is different from the lattice constant a0 = 0.25nm (Fig. 10b), and can be described by the electronic wavefunctions in the nanotube. Since dl/dV is a measure of the squared amplitude of the wavefunction, they were able to fit the experimental dl/dV with the trial function Asin2(2^x/A + + B. This enabled the separation between the dl/dV peaks to be correlated with half the Fermi wavelength AF. The calculated value for AF = 3a0 = 0.74 nm, determined from the two linear bands crossing at kF, is in good agreement with experimental observations. Hence discrete electron standing waves were observed in short armchair nanotubes. It is also worth noting that the observed widths of the nanotubes probed in these investigations, ~ 10 nm, are significantly larger than expected for a single SWNT. This suggests that it is likely the measurements were on ropes of SWNTs. In this regard, it will be important in the future to assess how tube-tube interactions perturb the quantum states in a single SWNT.

It is possible that additional features in the electronic structure of finite-sized nanotubes may appear in lengths nearly an order of magnitude shorter. To this end, Odom et al. [29] have studied quantum size effects in both chi-ral metallic and semiconducting tubes. STM images of nanotubes shortened to six and five nanometers, respectively, are shown in Fig. 11a,b. The I-V measurements show a step-wise increase of current over a two-volt bias range for both tubes, and the observed peaks in the (V/I)dI/dV were attributed to resonant tunneling through discrete energy levels resulting from the finite length of the SWNT. To first order, analysis of the peak spacing for

Bias Voltage (V)

Fig. 11. STM imaging and spectroscopy of finite-size SWNTs. (a—c) SWNTs cut by voltage pulses and shortened to lengths of 6 nm, 5 nm, and 3 nm, respectively. (d—f) Averaged normalized conductance and I-V measurements performed on the nanotubes in (a—c), respectively. Six I-V curves were taken along the tube length and averaged together since the spectra were essentially indistinguishable [29]

Bias Voltage (V)

Fig. 11. STM imaging and spectroscopy of finite-size SWNTs. (a—c) SWNTs cut by voltage pulses and shortened to lengths of 6 nm, 5 nm, and 3 nm, respectively. (d—f) Averaged normalized conductance and I-V measurements performed on the nanotubes in (a—c), respectively. Six I-V curves were taken along the tube length and averaged together since the spectra were essentially indistinguishable [29]

the finite-sized nanotubes (Fig. 11d,e) agrees with the simple 1D particle-in-a-box model. The former tube that is six nanometers long exhibits a mean peak spacing of approximately 0.27 eV. A six nanometer tube within this 1D box model would have an average level spacing A E ~ 1.67 eV/6 = 0.28 eV. For the latter tube with its shorter length, the observed peak spacing is also wider, as expected from this model.

In addition, an atomically resolved SWNT, only three nanometers long was investigated (Fig. 11c). The normalized conductance of this short piece shown in Fig. 11f appears, however, quite different from the expected 1.67 eV/3 = 0.55 eV energy level spacing for a nanotube three nanometers long. This is not surprising due to the limitations of this simple model, and the need for a more detailed molecular model to explain adequately the electronic structure is evident. Ab initio calculations of SWNT band structure have recently shown that the energy level spacing of finite-size tubes may be considerably different from that predicted from a Huckel model due to the asymmetry and shifting of the linear bands crossing at EF [48]. In addition, several molecular computational studies have predicted that nanotubes less than four nanometers long should open a HOMO-LUMO gap around Ef , although its magnitude varies greatly among different calculation meth ods [49,50]. These studies have been performed on finite-sized, open-ended, achiral (n, 0) zigzag and (n, n) armchair tubes. In quantum chemistry calculations, symmetry considerations are important, and in this regard, chiral nano-tubes may exhibit drastically different electronic characteristics compared with achiral ones. Clearly, more sophisticated molecular and first principle calculations are required to fully understand nanotubes at such ultra-short length scales.

In contrast to the metallic nanotubes and other nanoscale systems [51], no significant length dependence is observed in finite-sized semiconducting nano-tubes down to five nanometers long [52]. Namely, tunneling spectroscopy data obtained from the center of the shortened tube showed a striking resemblance to the spectrum observed before cutting. That is, the positions of the valence and conduction bands are nearly identical before and after cutting. Spectra taken at the ends also exhibited the same VHS positions, and a localized state near ~0.2eV, which is attributed to dangling bonds, was also observed. It is possible that long-length scale disorder and very short electron mean free paths (~2nm) in semiconducting tubes [53] may account for the similar electronic behavior observed in short and long nanotubes. This suggests that detailed studies should be carried out with even shorter tube lengths. However, recent ab initio calculations on the electronic structure of semiconducting nanotubes with lengths of two-three nanometers also seem robust to reduced-length effects and merely reproduce the major features of the bulk DOS (i.e. the energy gap) [54].

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